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Solving partial differential equations on (evolving) surfaces with radial basis functions

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Abstract

Meshfree, kernel-based spatial discretisations are recent tools to discretise partial differential equations on surfaces. The goals of this paper are to analyse and compare three different meshfree kernel-based methods for the spatial discretisation of semi-linear parabolic partial differential equations (PDEs) on surfaces, i.e. on smooth, compact, connected, orientable, and closed (d − 1)-dimensional submanifolds of \(\mathbb {R}^{d}\). The three different methods are collocation, the Galerkin, and the RBF-FD method, respectively. Their advantages and drawbacks are discussed, and previously known theoretical results are extended and numerically verified. Finally, a significant part of this paper is devoted to solving PDEs on evolving surfaces with RBF-FD, which has not been done previously.

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Authors and Affiliations

  1. Applied and Numerical Analysis, Department of Mathematics, University of Bayreuth, 95440, Bayreuth, Germany

    Holger Wendland & Jens Künemund

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  1. Holger Wendland
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  2. Jens Künemund
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Correspondence to Holger Wendland.

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Communicated by: Tobin Driscoll

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Wendland, H., Künemund, J. Solving partial differential equations on (evolving) surfaces with radial basis functions. Adv Comput Math 46, 64 (2020). https://doi.org/10.1007/s10444-020-09803-0

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